The relationship between a topological Yang-Mills field

and a magnetic monopole

The relationship between a topological Yang-Mills field

and a magnetic monopole

RCNP, Osaka, 7 Dec. 2010.

Nobuyuki Fukui (Chiba University, Japan)Kei-Ichi Kondo (Chiba University, Japan)

Akihiro Shibata (Computing Research Center, KEK, Japan)Toru Shinohara (Chiba University, Japan)

Based on N. Fukui, K.-I. Kondo, A. Shibata, T. Shinohara, Phys. Rev. D82 045015 (2010)

Contents• Introduction• Reformulation of SU(2) Yang-Mills theory• Numerical calculation• Results• Summary

1. Introduction

Magnetic monopoles are indispensable to quark confinement from aviewpoint of the dual superconductor picture.

We proposed a reformulation of Yang-Mills theory to extract the magnetic monopolefrom the theory with keeping gauge symmetry.K.-I. Kondo., T. Murakami and T. Shinohara, Prog. Theor. Phys. 115 201-216 (2006)K.-I. Kondo., T. Murakami and T. Shinohara, Prog. Theor. Phys. 120 1-50 (2008)

Our purpose is to show that the magnetic monopole existsin Yang-Mills theory.

In this talk, I show that the magnetic monopole comes out of instantonsolutions based on the reformulation numerically.

2. Reformulation of Yang-Mills theory

original Yang-Mills

enlarged Yang-Mills

Reduction condition

reformulated Yang-Mills

1. By introducing a color vector field with a unit length, We constructed “enlarged Yang-Mills” with the enlarged gauge symmetry .

2. We impose the reduction condition to reduce the enlarged gauge symmetry to ,

equipollent

The reduction condition and a definition of gauge-invariant magnetic monopole

Reduction functional

The reduction condition is given by minimizing the reduction functional under the enlarged gauge transformation:

The local minima are given by the differential equation which we call the reduction differential equation (RDE):

In the reformulated Yang-Mills, a composite field

is very important. The field strength of is parallel to :

So we can define the gauge-invariant field strength and the gauge-invariant monopole current as

3. Numerical calculation

We use a lattice regularization for numerical calculations.A link variable is computed by

The reduction functional on a lattice is given by

where is a unit color field on each site,

We introduce the Lagrange multiplier. Then, the stationary condition for the reduction functional is given by

After a little calculation, we obtain a lattice version of the RDE:

boundary

S. Ito, S. Kato, K.-I. Kondo, T. Murakami, A. Shibata and T. Shinohara, Phys. Lett. B645, 67-74 (2007).A. Shibata, K.-I. Kondo and T. Shinohara, Phys. Lett. B691, 91 (2010).

The V-part on a lattice is given by

The monopole current on a lattice is constructed as

We recall that the instanton configuration approaches a pure gauge at infinity:

So, we adopt a boundary condition as

Then, we assume that behaves asymptotically

A boundary condition

1. We calculate for

2. We solve

3.

the procedure of a numerical calculation

under the boundary condition.

4. Results

We calculate the magnetic monopole for• Regular one-instanton• Jackiw-Nohl-Rebbi (JNR) type two-instanton

Here, I give a detailed account of the result of JNR type two-instanton.

Jackiw-Nohl-Rebbi (JNR) type two-instanton

In this case, is

Hopf map

Consequently, boundary condition is

In the calculation, we equate three size parametersand put three pole positions on plane,so that the three poles are located atthe vertices of an equilateral triangle:

size pole

JNR two-instanton and the associated magnetic-monopole current for various choice of .The grid shows an instanton charge density on plane.These figure show that monopole currents form a circular loop. The circular loops are located on the plane specified by three poles of the JNR two-instanton.

The monopole current has a non-zero value on a small number of links.This table indicates that the size of the circular loop increases proportionally as r increases.

The relationship between and the magnetic-monopole loop

The configuration of the color field and a circular loop of the magnetic monopolecurrent obtained from the JNR two-instanton solution , viewed in (a) the plane which is off three poles, and (b) the plane which goes through a pole . Here the SU(2) color field is identified with a unit vector inthe three-dimensional space. These figure show the color vector field is winding around the loop, and it’sdirection is indeterminate on the loop.

conclusion

・ We show that the magnetic monopole comes out of an instanton. The magnetic monopole for the JNR two-instanton shapes a circular loop.

・ We found the relationship between the magnetic monopole and the singular point of the color field . is winding around the loop.

future problem

• solving the RDE analytically.

• computing contribution of the magnetic monopole configuration to physical quantities (ex. Wilson loop).

• extending to finite-temperature field theory

5. Summary

Thank you for your attention!Thank you for your attention!

Hopf map

In this case, is

and the boundary condition is

regular one-instanton (BPST type)

centersize

In the calculation, we fix the center on the origin

and change the value of size .

One instanton in the regular gauge and the associated magnetic-monopole current for various choice of size parameter .The grid shows an instanton charge density on plane.These figures show that non-zero monopole currents form a small loop.

The monopole current has a non-zero value on a small number of links.The size of the loop hardly changes while the size parameter increases.

A numerical technique

RDE on a lattice

・・・・・　 (A)

We recursively apply (A) to on each site and update it until converges.At this time, We fix on a boundary of a finite lattice.

1. We give a initial configuration for n.

2.

A. Shibata, K.-I. Kondo, S. Kato, S. Ito, T. Shinohara, and N. Fukui,Proceedings of the 27th International Symposium on Lattice Field Theory (Lattice 2009),Beijing, China, 2009　 (arXiv:0911.4533).

The monopole loop in a lattice simulation